Geoscience Reference
In-Depth Information
Iwagaki (
1956
) analyzed the equilibrium of a single spherical particle, placed on
a rough surface, and found the conditions necessary for the equilibrium of a particle
in different conditions of viscous sublayer. The theoretical equation given by him is
of the form:
'
e
s
C
s
R
tan
Y
c
¼
(15)
function of
R
.
The analysis of Ikeda (
1982
) that is based on Iwagaki (
1956
) and Coleman
(
1967
) could approximately derive the Shields diagram. The analysis was based
on forces acting on a solitary particle placed on a sediment bed. He obtained an
equation as follows:
e
s
¼
empirical coefficient for the sheltering effect; and
C
s
¼
where
(
)
0
:
6
10
=
3
4
3
tan
'
4
:
5
R
08
R
10
=
3
þ k
1
ln 1
Y
c
¼
C
L
Þ
10
:
þ
(16)
ð
C
D
þ
tan
'
1
þ
0
:
3
R
where
von K´rm´n constant.
On a horizontal bed, the expression for the force balance given by Wiberg and
Smith (
1987
) leads to:
k ¼
2
C
D
a
0
1
tan
'
Y
c
¼
z
0
Þ
(17)
f
2
ð
z
=
1
þð
F
L
=
F
D
Þ
c
tan
'
where
a
0
¼
A
x
d
/
V
;
A
x
¼
frontal area of the particle;
V
¼
volume of the particle;
(
C
L
/
C
D
)
f
2
(
z
/
z
0
)/
z
¼
elevation from the bed;
z
0
¼
zero-velocity level;
F
L
/
F
D
¼
[
f
2
(
z
T
/
z
0
)
f
2
(
z
B
/
z
0
)];
z
T
¼
elevation of the top point of the particle from the bed;
and
z
B
¼
elevation of the bottom point of the particle from the bed.
Dey (
1999
) and Dey and Papanicolaou (
2008
) analyzed the hydrodynamic forces
acting on a solitary particle resting over a horizontal sediment bed in a three-
dimensional configuration (Fig.
3
), including the effect of turbulent fluctuations.
z
F
L
u
D
z
0
F
D
Z
M
x
d
X
F
G
Fig. 3 Forces acting on a solitary particle in a three-dimensional configuration
